Notes on the Nonlinear Dependence of a Multiscale Coupled System with Respect to the Interface

NOTES ON THE NONLINEAR DEPENDENCE OF A MULTISCALE COUPLED SYSTEM WITH RESPECT TO THE INTERFACE Fernando A Morales Escuela de Matemáticas,Universidad Nacional de Colombia, Sede Medell´ın. Calle 59 A No 63-20, Of 43-106. Medell´ın,Colombia. Abstract. This work studies the dependence of the solution with respect to interface geometric perturbations in a multiscaled coupled Darcy flow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence are presented, as well as sufficient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case using orthogonal decompositions on appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multiple dimensional setting, studying the nonlinear operators introduced by the geometric perturbations. 1. Introduction The study of saturated flow in geological porous media frequently presents nat- ural structures with a dense network of fissures nested in the rock matrix [21]. It is also frequent to observe vuggy porous media [1], which have the presence of cavities in the rock matrix significantly larger than the average pore size of the medium. This is a multiple scale physics phenomenon, because there are regions of the medium where the flow velocity is significantly larger than the velocity on the other ones. The modeling of the interface between regions is subject to very active research: first, fluid transmission conditions across the interface are of great importance, see [19,3] for discussion of the governing laws; see [7,2] for a numerical point of view; see [1, 12,5, 16] for the analytic approach and [20,4] for a more general perspec- tive. Second, the placement of the interface is debatable since a boundary layer phenomenon between regions occurs, see [10, 18] for discussion. The interface cou- arXiv:1312.6912v3 [math.AP] 15 Mar 2014 ples regions of slow velocity (order O(1)) and fast flow (order O(1/)), see figure1. Hence, its placement and geometric description become an important issue because perturbations of the interface are inevitable. On one hand the geological strata data available are always limited, on the other hand the numerical implementation of models involving curvy interfaces, in most of the cases can only approximate the real surface. Finally, on a very different line, in the analysis of saturated flow in deformable porous media, one of the aspects is the understanding the geometric perturbations of an interface of reference. 1991 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Multiscale coupled systems, interface geometric perturbations, variational formulations, nonlinear dependence. The author was supported by the project HERMES 17194 from Universidad Nacional de Colombia. 1 2 FERNANDO A MORALES Figure 1. Original Domain and a Perturbation Ωζ 2 Ω = k 2 2 k ϵ = k 2 ζ k ϵ Γ Γ Γ = k k 1 k=k Ωζ 1 Ω 1 1 Clearly, the continuity of the solution with respect to the geometry of the interface is an important issue which has received very little treatment and mainly limited to flat interfaces. Most of the theoretical achievements in the field of multiscale coupled systems, concentrate their efforts in removing the singularities introduced by the scales using homogenization processes. These techniques can be either formal [17, 13], analytic [11] or numerical [14]. Here, we model the stationary problem with a coupled system of partial differen- tial equations of Darcy flow in both regions, in direct variational formulation. We 1 simulate the region of fast flow scaling by the ratio of permeability over viscos- ity, as in figure1. It will be assumed that the real interface Γ is horizontal flat and the perturbed one Γζ is curved. Of course a flat surface will not be perturbed when discretized and seems unrealistic to consider perturbations of it. Our choice is motivated by two reasons: first, for the sake of clarity in the notation, calcu- lation and interpretation of the results. Second, when studying the phenomenon of saturated flow in deformable porous media perturbations of a flat surface are of interest. Finally it is important to highlight that the mathematical essentials of the problem are captured in this framework. The paper starts proving in section 2, that the solutions depend continuously with respect to the interface, then it moves to the much deeper question of exploring the dependence itself. In section 3 the one dimensional case the rate of convergence question is solved completely using orthogonal decomposition of adequate subspaces, finally section 4 reveals the highly nonlinear dependence of the solutions with respect to the interface. We close this section introducing the notation. Vectors are denoted by boldface letters, as are vector-valued functions and corresponding function spaces. We use xe to indicate a vector in RN−1; if x 2 RN then the RN−1 ×f0g projection is identified def with xe = (x1; x2; : : : ; xN−1) so that x = (xe; xN ). The symbol re represents the RN R R gradient in the first N − 1 derivatives. Given a function f : ! then M f dS RN−1 RN R is the notation for its surface integral on the manifold M ⊆ . A f dx stands for the volume integral in the set A ⊆ RN ; whenever the context is clear INTERFACE PERTURBATIONS AND NON-LINEARITY 3 R 2 we simply write A f. The notation k · k0;A, k · k1;A respectively denote the L (A) 1 N and H (A) norms on the domain A ⊆ R . 1A stands for the indicator function of any given set A. The be` indicates the unitary vector in the `-th direction for RN 1 ≤ ` ≤ N. We denote by νb the outwards normal vector to smooth domain in and n indicates the upwards normal vector i.e. n · eN ≥ 0. Finally, the Lebesgue b N b b measure in R is denoted by λN . 2. Formulation and Convergence 2.1. Geometric Setting. In the following Γ denotes a connected set in RN−1×f0g whose projection onto RN−1 is open; from now on we make no distinction between these two domains. Similarly, Ω 1; Ω 2 denote be smooth bounded open regions in RN i separated by Γ, i.e. @Ω 1 \ @Ω 2 = Γ; and such that sgn(x · ebN ) = (−1) for each x 2 Ωi, i = 1; 2 (see figure1). Next, we introduce the admissible perturbations of the interface Γ. Definition 2.1. We say the set T (Γ; Ω) of piecewise C1 perturbations of the interface Γ contained in Ω is given by def (1) T (Γ; Ω) = fζ 2 C Γ :(xe; ζ(xe )) 2 Ω 8 xe 2 Γ 1 ; ζ j@Γ = 0 and ζ is a piecesise C function g: The interface associated to ζ 2 T (Γ; Ω) is given by the set ζ def RN (2a) Γ = f(xe; ζ(xe )) 2 : xe 2 Γg: The domains associated to ζ 2 T (Γ; Ω) are defined by the sets ζ def (2b) Ω 1 = f(xe; xN ) 2 Ω: xe 2 Γ; xN < ζ(xe )g ζ def (2c) Ω 2 = f(xe; xN ) 2 Ω: xe 2 Γ; ζ(xe ) < xN g Remark 1. Observe the following facts ζ ζ ζ (3a) @Ω 1 \ @Ω 2 = Γ ; ζ ζ ζ (3b) Ω 1 [ Γ [ Ω 2 = Ω; ζ ζ (3c) @Ω 1 − Γ = @Ω 1 − Γ; ζ ζ (3d) @Ω 2 − Γ = @Ω 2 − Γ: Definition 2.2. Define the space def 1 (4) V = fu 2 H (Ω) : uj @Ω1−Γ = 0g Endowed with the inner product h·; ·i : V × V ! R def Z (5) hu; viV = ru · rv; Ω def p and the norm kukV = hu; uiV . Remark 2. Recall that due to the boundary condition defining the space V and 1 the Poincaréinequality the k · kV -norm is equivalent to the standard H -norm. 4 FERNANDO A MORALES 2.2. The Problems. Consider the strong problem k (6a) − r · i rp = F in Ω ; i = 1; 2; i−1 i i with the interface conditions k (6b) p = p ; k rp · n − 2 rp · n = f on Γ; 1 2 1 1 b 2 b and the boundary conditions (6c) p1 = 0 on @Ω1 − Γ; rp2 · nb = 0 on @Ω2 − Γ: Now, its perturbation in strong form is given by k (7a) − r · i rq = F in Ωζ ; i = 1; 2; i−1 i i with the interface conditions k (7b) q = q ; k rq · n − 2 rq · n = f on Γ ζ ; 1 2 1 1 b 2 b and the boundary conditions ζ ζ ζ ζ (7c) q1 = 0 on @Ω1 − Γ ; rq2 · nb = 0 on @Ω2 − Γ : Both systems above model stationary Darcy flow, coupling the regions depicted in the left and right hand side of figure1 respectively. The coefficients k1, k2 indicate the permeability in the corresponding domain; for simplicity they will be omitted 1 1 in the following. The scaling factor ensures a much higher velocity O( ) in the upper region with respect to the lower region fluid velocity O(1). The term F stands for fluid sources and f for a normal flux forcing term on the interface; it is assumed that f is well defined L2-function on both manifolds Γ and Γζ . Hence, the weak problems in direct formulation are given by Z 1 Z Z Z (8a) p 2 V : rp · rr + rp · rr = F r + f r d S ; 8 r 2 V Ω1 Ω2 Ω Γ Z 1 Z Z Z (8b) q ζ 2 V : rq ζ · rr + rq ζ · rr = F r + f r d S ; 8 r 2 V ζ ζ ζ Ω1 Ω2 Ω Γ Theorem 2.3.

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